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Designing spherical microphone arrays using periodic tilngs on the sphere

There is a small but important difference in the design of spherical microphone and speaker arrays. In both cases we want to use sphere efficiently by "evenly" distributing the transducers. In the loudspeaker situation we have the additional constraint that we want to maximize the available transducer surface area to get the most energy (and in the case of sound waves to get as much low frequency energy as we can). So in the speaker case we have been drawn to apply the theory of circular disc packing on spheres. For examples of this approach planar wavefield applications and for spherical speakers take a look at my notes here:http://cnmat.berkeley.edu/user/adrian_freed/blog/2007/10/11/using_period...

"This sections describes sampling schemes where the samples
are distributed uniformly and nearly uniformly on the surface
of a sphere. A sampling scheme where the distance between
neighboring samples is constant, give rise to a limited set of
special geometries called platonic solids, e.g., 20 samples on
the vertices of a dodecahedron. A sampling scheme with
the samples distributed nearly uniformly on the sphere surface
offer a much wider range of configurations. Hardin and Sloane
prove that the number of samples for these configurations is
at least (n+1)^2, but show that in practice this number is larger
than 1.5(n+1)^2 for many examples, when the weights are
taken to be equal. Lebedev also provides samples and weights for various
configurations, with a total of about 1.3(n+2^2samples."

The problem with both the near uniform sampling and disk packing approaches is that optimal placement has no transducers at equiangle locations. In other words, they are expensive to build because however we choose to break up the sphere to assemble it the transducer locations are unique on each facet. The approach I took for our icosahedral speaker geometry was to compromise the optimal placement in favor of something buildable on planar facets. The particular compromise of six close packed speakers on 20 faces matches implementation constraints quite well but leaves about 10% of the available surface area of the sphere void of transducers. This may be addressed by plugging the 12 pentangle vertex holes with drivers. The disk packing however is so efficient at our current diameter that it is hard to find room for pole mounts, suspension hooks and cables. Among the numerous applications (and control structure models) of our speaker the one that is probably most effected by our compromise is beamforming. Since this is one of the most important applications of microphone arrays mirroring our speaker designs will not be a good idea. For the same engineering effort better performance can be achieved for microphones.

NB: the following represents ideas which may be potentially patentable and should therefor not be made public:

1) Single planar face shape
The first designs to explore involve the constraint that only one planar shape be used to build the sphere from and that the pattern of microphones be the same on each facet. Let's start considering the platonic solids. The larger ones (icosahedron and dodecahedron) have the advantage over the cube for example that the edges (where transducers cannot be placed) are shortest. This is one way of thinking about how more freedom will be available to approximate the optimal transducer placement on each face. There is another constraint we want to consider. We would like symmetry in the placement on each face because it simplifies the signal processing. On the speaker array there are only two equivalence classes of transducers: the triangular face vertex ones and the edge ones.

Judging by the microphone arrays built so far it has been hard for people to move away from the platonic solids and their derivatives. I admit that it took me an embarassingly long time to realize that when the transducer count goes up we can abandon our reliance on those shapes. One important family to consider is the Catalan solids:

The 30 faces are rhombus shaped and more efficient and easier to employ from a PCB layout point of view than triangles. 4 microphones on each face create a 120 transducer array and the microphones can be arranged in pairs with symmetry and just 2 equivalence classes from the signal processing point of view. Note that this shape can be readily split in two like an egg-shell (as we do for our speaker) and would have less mechanical interference issues than an icosahedron because the angle between faces is smaller. Small stereo ADC chips are a natural match to this arrangement and it is compatible with our dual-board approach to the communications controller.

Some of the other Catalan solids are relevant but probably only useful for larger arrays. The pentakis dodecahedron with three transducers per triangular face yields a 180 transducer array. The hexakis octahedron yields 144. The deltoidal hexecontahedron would result in a 120 transducer array with 2 microphones/face. It is usually better to have more transducers/face so that the economies of scale lower the number of cables to the internal controller boards.

2) Unit Construction

The idea here is to build the sphere from fewer solid pieces by combining 2 or more planar faces. There are geometric and practical limits to this approach. Too many combined faces and we cannot tile the sphere without introducing new shapes. For example on the icosahedron we can tile it with units of two adjacent faces. Since 3 doesn't divide 20 we cant build with a single three-triangle unit. The rhombic triacontahedron can be build from 10 3-unit tiles. This is comparable to the number of (easier) castings required for a dodecahedron but simulations may show that the extra complexity may be worth it in terms of beam forming ability. Note that the we could go to 5 strips of 6-unit tiles (and 5 strips of 4-unit tiles for the icosahedron). Even if these longer strips are impractical in a single casting it is important to understand the geometries because they suggest a way to simplify the cabling considerably: short compact clutter-free board-board connections along the strip and the just a few connections from the strips to the central controller.

A practical constraint to unit construction involves avoiding undercuts in the casting of the units. I view unit construction as a second step in the engineering optimization to explore once the core geometry is decided. Unit construction of polyhedra has been explored in unit origami texts.

The truncated icosahedron is a logical starting point for a 120 array by distributing 5 microphones symmetrically near the edges of the 12 pentagons and 3 near the pentangle joining edges of the 20 hexagons. With a unit built from two adjacent hexagons the circuit boards for each face shape would have a comparable number of microphones and the total number of units is 22: comparable to the icosahedron (20).

The snub cube is a basis for a 3*32 +4 *6 = 120 array and its 6 square faces suggest additional design freedoms for
mounting, connectivity or controller electronics.

The small rhombicuboctahedron suggests a 3*8 +4*18 = 96 array with the advantage of lots of square pcb's

The Archimean solids with 3 face shapes dont seem to be worth the trouble and the namber of shapes is hard to reduce with the unit construction technique